Biophotonics_Concepts_to_Applications

sphere size, but a nominal value is around 50 pN/μm. After the trap stiffness has
been determined, then the force on a particle in the trap can be determined by
measuring the position of the bead in the trap.

Example 11.2Consider an optical trap that has a stiffness k = 50 pN/μm. If

an image analysis instrument can determine the position of a

micrometer-sized sphere to within 10 nm, what is the force resolution?

Solution: From Eq. (11.1) the force resolution is

F¼kx¼ðÞ50 pN=lmðÞ¼10 nm 0 :5pN:

For small particles the gradient force can be expressed in terms of the gradient of
the intensity∇Ias

F

!

grad¼

2 pnmedr^3

c

m^2  1

m^2 þ 2



rI

!

ð 11 : 2 Þ

where r is the radius of the particle, c is the speed of light, nmedis the refractive
index of the surrounding medium, npartis the refractive index of the particle, and
m=npart/nmedis the ratio of the refractive indices of the particle and the sur-
rounding medium, respectively. The magnitude of the scattering force is

Fscat¼

8 pnmedr^6

3c

2 p

k

 4

m^2  1

m^2 þ 2



I ð 11 : 3 Þ

For a power P incident on the surface of the sphere, the total force Ftotalexerted
on a particle is the sum of the force contributions due to the scattered reflected rays
and the forces resulting from the emerging refracted rays. This total force can be
expressed as

Ftotal¼Q

nmedP

c

ð 11 : 4 Þ

Here, nmedis the refractive index of the medium in which the sphere is located.
Thetrapping efficiencyorquality factorQ depends on the Fresnel reflection and
transmission coefficients R and T, respectively, and gives a measure of the
momentum that the light beam transferred to the sphere. The detailed calculations
of Q can be found in the literature [ 1 , 2 ]. The value of Q depends on the type of
optical trap, the NA of the microscope objective, and the size of the trapped particle.
Optical tweezers can have trapping efficiencies up to Q = 0.3 perpendicular to the
direction of the laser beam propagation.

328 11 Biophotonics Technology Applications